Singular Weyl's law with Ricci curvature bounded below

Abstract

We establish two surprising types of Weyl's laws for some compact $\mathrm{RCD}(K,N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for $\mathrm{RCD}(K,N)$ spaces. Our results depends crucially on analyzing and developing important properties of the examples constructed by the last two authors, showing them isometric to the $\alpha$-Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures by Cheeger-Colding and by Kapovitch-Kell-Ketterer.

Final version. To appear in Trans. AMS Series B. 41 pages

References in corpus (5)

• 100 years of Weyl's law
• Examples of Ricci limit spaces with non-integer Hausdorff dimension
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• The Grushin hemisphere as a Ricci limit space with curvature $\ge 1$
• The Li-Yau Inequality and Heat Kernels on Metric Measure Spaces

Cited by in corpus (1)

• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature